Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C8 · Schur multiplier (H2) = C23 |
| direct product | C7×#11586 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C8 | C22⋊C4 |
| commutator | C2×C4 | C22×C7×C8 |
| Frattini, cosocle | C4×C8 | C23×C7 |
| socle | C22×C7 | C2×C22⋊C8 |
| prime radical | C7×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C8 ◁ C2×C4×C7×C8 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,19,24,21,25,32,39,20,23,31,36,33,37,44,46)(18,26,28,38,29,40,42,47,27,22,30,34,41,35,43,45);(9,12)(10,15)(13,14)(16,18)(17,22)(19,30)(20,27)(21,29)(23,26)(24,38)(25,35)(28,31)(32,43)(33,41)(34,36)(37,40)(39,47)(42,44)(45,46);(18,27)(22,26)(28,30)(29,41)(34,38)(35,40)(42,43)(45,47)532 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 4574);